Danny Ross Lunsford
Apr17-04, 05:08 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>I have some PDEs to solve.\n\nHere they are (make adjustments for notation):\n\nR{mn} = (2R/W) Tmn - (1/2W)(DmDn + DnDm) W\n\n1/S @n S R Fmn = 5/4 Dm W\n\n====================\n\nExplanation:\n\nIndic es go 1..6. The flat space metric is (---+++) that is, space-like\nis -, timelike is +.\n\nThe space is equipped with a symmetric connection,\n\nCmn,p = 1/2 (gmp,n + gnp,m - gnm,p) + 1/2 (gmp An + gnp Am - gmn Ap)\n\nIn addition to general covariance, there is a gauge invariance\n\ngmn -> L(x) gmn\n\nAm -> Am - @m log L(x) = Am - 1/L @m L\n\nThe connection is invariant under these gauge transformations.\n\nA tensor has a weight w under a gauge transformation:\n\nTab..c -> L^w Tab..c\n\nIn particular, gmn with lower mn is weight 1, and gmn with upper mn is\nweight -1. Am is weight 0.\n\nThere is a curvature tensor defined in the usual way having weight 0\n(because the connection is gauge invariant).\n\nThe ordinary covariant derivative is defined as usual:\n\ndm = @m + C..T... - C..T...\n\nwith + signs for each contravariant index, - signs for each covariant index.\n\nThe conformal covariant derivative of a tensor of weight w is\n\nDm T(w) = dm T(w) + w Am T(w)\n\nin particular for the metric\n\nDm gab = 0\n\nso\n\ndm gab = +-Am gab\n\nwith - sign for covariant g, + for contravariant.\n\nBasic weight rule: Raising an index decreases the weight by 1, lowering\nan index increases it by 1.\n\n============================================ =====\n\nNow to the equations:\n\nFmn = @m An - @n Am (twice covariant, weight 0)\n\nW = FmnFmn (weight -2)\n\nS = sqrt(det(gmn)) (weight 3)\n\nTmn = FmpFpn + 1/4 gmn W (weight -1)\n\nR{mn} is the symmetric part of the Ricci tensor (weight 0) and R is the\nRicci scalar, which involves terms in Am as follows:\n\nR = tr R{mn} = P + 5 1/S @m S Am + 5 AmAm\n\nwhere P involves only terms in gab. R is weight -1.\n\nNote there are identities\n\nDmDm W = 0\n\nDm(Rmn + 1/2 gmn R - 1/2 Fmn) = 0\n\n=====================================\n\nWith this information, find the spherically symmetric solution (in space\n*and* time, and in space alone with assumed "flat" time) that goes over\nto a flat space metric at infinity with free waves in Am. Verify the\nbehavior under gauge transformation. Assume R/W -> -4 pi G there (G the\ngravitational constant).\n\nNote: by spherically symmetric in space and time, we mean the metric is\ndefined by\n\ngab dxa dxb = M(r,s) {dt}^2 - N(r,s) {dx}^2\n\nwhere {dt}^2 is the "timespace" expressed in polar coordinates\ns,omega,psi and {dx}^2 is the "spacespace" expressed in polar\ncoordinates r,theta,phi.\n\nMore ambitious: find the general static solution with pseudo-spherical\nsymmetry in the remaining 5 coordinates (everything independent of\nx4=t). This is the direct analog of the Schwarzschild solution.\n\nIf you think the GR equations are hard, think again :) On the other\nhand, by the nature of things there are no sources here.\n\nBefore publishing my solution I\'d like to see what other people come up\nwith!\n\n-drl\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I have some PDEs to solve.
Here they are (make adjustments for notation):
R{mn} = (2R/W) Tmn - (1/2W)(DmDn + DnDm) W
1/S @n S R[/itex] Fmn = 5/4 Dm W
====================
Explanation:
Indices go 1..6. The flat space metric is (---+++) that is, space-like
is -, timelike is +.
The space is equipped with a symmetric connection,
Cmn,p = 1/2 (gmp,n + gnp,m - gnm,p) + 1/2 (gmp An + gnp Am - gmn Ap)
In addition to general covariance, there is a gauge invariance
gmn -> L(x) gmn
Am -> Am - @m log L(x) = Am - 1/L @m L
The connection is invariant under these gauge transformations.
A tensor has a weight w under a gauge transformation:
Tab..c -> L^w Tab..c
In particular, gmn with lower mn is weight 1, and gmn with upper mn is
weight -1. Am is weight .
There is a curvature tensor defined in the usual way having weight
(because the connection is gauge invariant).
The ordinary covariant derivative is defined as usual:
dm = @m + C..T... - C..T...
with + signs for each contravariant index, - signs for each covariant index.
The conformal covariant derivative of a tensor of weight w is
Dm T(w) = dm T(w) + w Am T(w)
in particular for the metric
Dm gab =
so
dm gab = +-Am gab
with - sign for covariant g, + for contravariant.
Basic weight rule: Raising an index decreases the weight by 1, lowering
an index increases it by 1.
=================================================
Now to the equations:
Fmn = @m An - @n Am (twice covariant, weight 0)
W = FmnFmn (weight -2)
S = \sqrt(det(gmn)) (weight 3)
Tmn = FmpFpn + 1/4 gmn W (weight -1)
R{mn} is the symmetric part of the Ricci tensor (weight 0) and R is the
Ricci scalar, which involves terms in Am as follows:
R = tr R{mn} = P + 5 1/S @m S Am + 5 AmAm
where P involves only terms in gab. R is weight -1.
Note there are identities
DmDm W =
Dm(Rmn + 1/2 gmn R - 1/2 Fmn) =
=====================================
With this information, find the spherically symmetric solution (in space
*and* time, and in space alone with assumed "flat" time) that goes over
to a flat space metric at infinity with free waves in Am. Verify the
behavior under gauge transformation. Assume R/W -> -4 \pi G there (G the
gravitational constant).
Note: by spherically symmetric in space and time, we mean the metric is
defined by
gab dxa dxb [itex]= M(r,s) {dt}^2 - N(r,s) {dx}^2
where {dt}^2 is the "timespace" expressed in polar coordinates
s,\omega,\psi and {dx}^2 is the "spacespace" expressed in polar
coordinates r,\theta,\phi.
More ambitious: find the general static solution with pseudo-spherical
symmetry in the remaining 5 coordinates (everything independent of
x4=t). This is the direct analog of the Schwarzschild solution.
If you think the GR equations are hard, think again :) On the other
hand, by the nature of things there are no sources here.
Before publishing my solution I'd like to see what other people come up
with!
-drl
Here they are (make adjustments for notation):
R{mn} = (2R/W) Tmn - (1/2W)(DmDn + DnDm) W
1/S @n S R[/itex] Fmn = 5/4 Dm W
====================
Explanation:
Indices go 1..6. The flat space metric is (---+++) that is, space-like
is -, timelike is +.
The space is equipped with a symmetric connection,
Cmn,p = 1/2 (gmp,n + gnp,m - gnm,p) + 1/2 (gmp An + gnp Am - gmn Ap)
In addition to general covariance, there is a gauge invariance
gmn -> L(x) gmn
Am -> Am - @m log L(x) = Am - 1/L @m L
The connection is invariant under these gauge transformations.
A tensor has a weight w under a gauge transformation:
Tab..c -> L^w Tab..c
In particular, gmn with lower mn is weight 1, and gmn with upper mn is
weight -1. Am is weight .
There is a curvature tensor defined in the usual way having weight
(because the connection is gauge invariant).
The ordinary covariant derivative is defined as usual:
dm = @m + C..T... - C..T...
with + signs for each contravariant index, - signs for each covariant index.
The conformal covariant derivative of a tensor of weight w is
Dm T(w) = dm T(w) + w Am T(w)
in particular for the metric
Dm gab =
so
dm gab = +-Am gab
with - sign for covariant g, + for contravariant.
Basic weight rule: Raising an index decreases the weight by 1, lowering
an index increases it by 1.
=================================================
Now to the equations:
Fmn = @m An - @n Am (twice covariant, weight 0)
W = FmnFmn (weight -2)
S = \sqrt(det(gmn)) (weight 3)
Tmn = FmpFpn + 1/4 gmn W (weight -1)
R{mn} is the symmetric part of the Ricci tensor (weight 0) and R is the
Ricci scalar, which involves terms in Am as follows:
R = tr R{mn} = P + 5 1/S @m S Am + 5 AmAm
where P involves only terms in gab. R is weight -1.
Note there are identities
DmDm W =
Dm(Rmn + 1/2 gmn R - 1/2 Fmn) =
=====================================
With this information, find the spherically symmetric solution (in space
*and* time, and in space alone with assumed "flat" time) that goes over
to a flat space metric at infinity with free waves in Am. Verify the
behavior under gauge transformation. Assume R/W -> -4 \pi G there (G the
gravitational constant).
Note: by spherically symmetric in space and time, we mean the metric is
defined by
gab dxa dxb [itex]= M(r,s) {dt}^2 - N(r,s) {dx}^2
where {dt}^2 is the "timespace" expressed in polar coordinates
s,\omega,\psi and {dx}^2 is the "spacespace" expressed in polar
coordinates r,\theta,\phi.
More ambitious: find the general static solution with pseudo-spherical
symmetry in the remaining 5 coordinates (everything independent of
x4=t). This is the direct analog of the Schwarzschild solution.
If you think the GR equations are hard, think again :) On the other
hand, by the nature of things there are no sources here.
Before publishing my solution I'd like to see what other people come up
with!
-drl